Two-dimensional rigid bodies in the $$xy$$ plane have three degrees of freedom. Position can be characterized by the $$x$$ and $$y$$ coordinates of a point on the object, and orientation by angle $$\theta_z$$ about an axis perpendicular to the plane. The complete movement of the body can be defined by two linear displacements $$\Delta x$$ and $$\Delta y\text{,}$$ and one angular displacement $$\Delta \theta_z\text{.}$$
Three-dimensional rigid bodies have six degrees of freedom, which can be specified with three orthogonal coordinates $$x, y$$ and $$z\text{,}$$ and three angles of rotation, $$\theta_x, \theta_y$$ and $$\theta_z\text{.}$$ Movement of the body is defined by three translations $$\Delta x\text{,}$$ $$\Delta y$$ and $$\Delta z\text{,}$$ and three rotations $$\Delta \theta_x\text{,}$$ $$\Delta \theta_y$$ and $$\theta_z\text{.}$$